Polynomials with height 1 and prescribed vanishing at 1. (English) Zbl 0999.12001

Borwein, Peter; Mossinghoff, Michael J.

Exp. Math. 9, No. 3, 425-433 (2000).

Given a positive integer \(m\), what are the real polynomials of minimal degree \(d(m)\) with all coefficients equal to one of \(0, \pm 1\) and divisible by \((x-1)^m\)? This paper describes two strategies for determining an upper bound on \(d(m)\) and presents various results on when such polynomials are divisible by certain cyclotomic polynomials of order \(p\) where \(p\) is a prime or equal to 4. A search algorithm that produces polynomials with the required coefficients of given degree \(d\) for given \(m\) is described and used to find the best examples for \(3 \leq m \leq 10\) and partial results when \(m = 11, 12\). For \(1 \leq m \leq 21\), the known polynomials of least degree are all products of polynomials of the type \(x^e - 1\). Three questions are posed. Are there polynomials of degree \(d(m)\) some of which are not of this type? Is there always a reciprocal polynomial of degree \(d(m)\)? Is \(d(21) < 294\)?

Reviewer: Edward J.Barbeau (Toronto)

MSC:

12D05	Polynomials in real and complex fields: factorization
12Y05	Computational aspects of field theory and polynomials (MSC2010)
11R09	Polynomials (irreducibility, etc.)

Keywords:

polynomials; height; zeros of high order; divisibility by cyclotomic polynomials; search algorithm

Software:

NTL

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